Precalculus Examples

\frac{3 x^{3} - 2 x^{2} + 3 x - 4}{x - 3}

$\frac{3 x^{3} - 2 x^{2} + 3 x - 4}{x - 3}$

Step 1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of

0

$0$ .

x

$x$

3

$3$

3 x^{3}

$3 x^{3}$

2 x^{2}

$2 x^{2}$

3 x

$3 x$

4

$4$

Step 2

Divide the highest order term in the dividend

3 x^{3}

$3 x^{3}$ by the highest order term in divisor

x

$x$ .

					$3 x^{2}$ $3 x^{2}$
	$x$ $x$	-	$3$ $3$		$3 x^{3}$ $3 x^{3}$	-	$2 x^{2}$ $2 x^{2}$	+	$3 x$ $3 x$	-	$4$ $4$

Step 3

Multiply the new quotient term by the divisor.

				$3 x^{2}$ $3 x^{2}$
$x$ $x$	-	$3$ $3$		$3 x^{3}$ $3 x^{3}$	-	$2 x^{2}$ $2 x^{2}$	+	$3 x$ $3 x$	-	$4$ $4$
			+	$3 x^{3}$ $3 x^{3}$	-	$9 x^{2}$ $9 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in

3 x^{3} - 9 x^{2}

$3 x^{3} - 9 x^{2}$

				$3 x^{2}$ $3 x^{2}$
$x$ $x$	-	$3$ $3$		$3 x^{3}$ $3 x^{3}$	-	$2 x^{2}$ $2 x^{2}$	+	$3 x$ $3 x$	-	$4$ $4$
			-	$3 x^{3}$ $3 x^{3}$	+	$9 x^{2}$ $9 x^{2}$

Step 5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$3 x^{2}$ $3 x^{2}$
$x$ $x$	-	$3$ $3$		$3 x^{3}$ $3 x^{3}$	-	$2 x^{2}$ $2 x^{2}$	+	$3 x$ $3 x$	-	$4$ $4$
			-	$3 x^{3}$ $3 x^{3}$	+	$9 x^{2}$ $9 x^{2}$
					+	$7 x^{2}$ $7 x^{2}$

Step 6

Pull the next terms from the original dividend down into the current dividend.

				$3 x^{2}$ $3 x^{2}$
$x$ $x$	-	$3$ $3$		$3 x^{3}$ $3 x^{3}$	-	$2 x^{2}$ $2 x^{2}$	+	$3 x$ $3 x$	-	$4$ $4$
			-	$3 x^{3}$ $3 x^{3}$	+	$9 x^{2}$ $9 x^{2}$
					+	$7 x^{2}$ $7 x^{2}$	+	$3 x$ $3 x$

Step 7

Divide the highest order term in the dividend

7 x^{2}

$7 x^{2}$ by the highest order term in divisor

x

$x$ .

				$3 x^{2}$ $3 x^{2}$	+	$7 x$ $7 x$
$x$ $x$	-	$3$ $3$		$3 x^{3}$ $3 x^{3}$	-	$2 x^{2}$ $2 x^{2}$	+	$3 x$ $3 x$	-	$4$ $4$
			-	$3 x^{3}$ $3 x^{3}$	+	$9 x^{2}$ $9 x^{2}$
					+	$7 x^{2}$ $7 x^{2}$	+	$3 x$ $3 x$

Step 8

Multiply the new quotient term by the divisor.

				$3 x^{2}$ $3 x^{2}$	+	$7 x$ $7 x$
$x$ $x$	-	$3$ $3$		$3 x^{3}$ $3 x^{3}$	-	$2 x^{2}$ $2 x^{2}$	+	$3 x$ $3 x$	-	$4$ $4$
			-	$3 x^{3}$ $3 x^{3}$	+	$9 x^{2}$ $9 x^{2}$
					+	$7 x^{2}$ $7 x^{2}$	+	$3 x$ $3 x$
					+	$7 x^{2}$ $7 x^{2}$	-	$21 x$ $21 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in

7 x^{2} - 21 x

$7 x^{2} - 21 x$

				$3 x^{2}$ $3 x^{2}$	+	$7 x$ $7 x$
$x$ $x$	-	$3$ $3$		$3 x^{3}$ $3 x^{3}$	-	$2 x^{2}$ $2 x^{2}$	+	$3 x$ $3 x$	-	$4$ $4$
			-	$3 x^{3}$ $3 x^{3}$	+	$9 x^{2}$ $9 x^{2}$
					+	$7 x^{2}$ $7 x^{2}$	+	$3 x$ $3 x$
					-	$7 x^{2}$ $7 x^{2}$	+	$21 x$ $21 x$

Step 10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$3 x^{2}$ $3 x^{2}$	+	$7 x$ $7 x$
$x$ $x$	-	$3$ $3$		$3 x^{3}$ $3 x^{3}$	-	$2 x^{2}$ $2 x^{2}$	+	$3 x$ $3 x$	-	$4$ $4$
			-	$3 x^{3}$ $3 x^{3}$	+	$9 x^{2}$ $9 x^{2}$
					+	$7 x^{2}$ $7 x^{2}$	+	$3 x$ $3 x$
					-	$7 x^{2}$ $7 x^{2}$	+	$21 x$ $21 x$
							+	$24 x$ $24 x$

Step 11

Pull the next terms from the original dividend down into the current dividend.

				$3 x^{2}$ $3 x^{2}$	+	$7 x$ $7 x$
$x$ $x$	-	$3$ $3$		$3 x^{3}$ $3 x^{3}$	-	$2 x^{2}$ $2 x^{2}$	+	$3 x$ $3 x$	-	$4$ $4$
			-	$3 x^{3}$ $3 x^{3}$	+	$9 x^{2}$ $9 x^{2}$
					+	$7 x^{2}$ $7 x^{2}$	+	$3 x$ $3 x$
					-	$7 x^{2}$ $7 x^{2}$	+	$21 x$ $21 x$
							+	$24 x$ $24 x$	-	$4$ $4$

Step 12

Divide the highest order term in the dividend

24 x

$24 x$ by the highest order term in divisor

x

$x$ .

				$3 x^{2}$ $3 x^{2}$	+	$7 x$ $7 x$	+	$24$ $24$
$x$ $x$	-	$3$ $3$		$3 x^{3}$ $3 x^{3}$	-	$2 x^{2}$ $2 x^{2}$	+	$3 x$ $3 x$	-	$4$ $4$
			-	$3 x^{3}$ $3 x^{3}$	+	$9 x^{2}$ $9 x^{2}$
					+	$7 x^{2}$ $7 x^{2}$	+	$3 x$ $3 x$
					-	$7 x^{2}$ $7 x^{2}$	+	$21 x$ $21 x$
							+	$24 x$ $24 x$	-	$4$ $4$

Step 13

Multiply the new quotient term by the divisor.

				$3 x^{2}$ $3 x^{2}$	+	$7 x$ $7 x$	+	$24$ $24$
$x$ $x$	-	$3$ $3$		$3 x^{3}$ $3 x^{3}$	-	$2 x^{2}$ $2 x^{2}$	+	$3 x$ $3 x$	-	$4$ $4$
			-	$3 x^{3}$ $3 x^{3}$	+	$9 x^{2}$ $9 x^{2}$
					+	$7 x^{2}$ $7 x^{2}$	+	$3 x$ $3 x$
					-	$7 x^{2}$ $7 x^{2}$	+	$21 x$ $21 x$
							+	$24 x$ $24 x$	-	$4$ $4$
							+	$24 x$ $24 x$	-	$72$ $72$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in

24 x - 72

$24 x - 72$

				$3 x^{2}$ $3 x^{2}$	+	$7 x$ $7 x$	+	$24$ $24$
$x$ $x$	-	$3$ $3$		$3 x^{3}$ $3 x^{3}$	-	$2 x^{2}$ $2 x^{2}$	+	$3 x$ $3 x$	-	$4$ $4$
			-	$3 x^{3}$ $3 x^{3}$	+	$9 x^{2}$ $9 x^{2}$
					+	$7 x^{2}$ $7 x^{2}$	+	$3 x$ $3 x$
					-	$7 x^{2}$ $7 x^{2}$	+	$21 x$ $21 x$
							+	$24 x$ $24 x$	-	$4$ $4$
							-	$24 x$ $24 x$	+	$72$ $72$

Step 15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$3 x^{2}$ $3 x^{2}$	+	$7 x$ $7 x$	+	$24$ $24$
$x$ $x$	-	$3$ $3$		$3 x^{3}$ $3 x^{3}$	-	$2 x^{2}$ $2 x^{2}$	+	$3 x$ $3 x$	-	$4$ $4$
			-	$3 x^{3}$ $3 x^{3}$	+	$9 x^{2}$ $9 x^{2}$
					+	$7 x^{2}$ $7 x^{2}$	+	$3 x$ $3 x$
					-	$7 x^{2}$ $7 x^{2}$	+	$21 x$ $21 x$
							+	$24 x$ $24 x$	-	$4$ $4$
							-	$24 x$ $24 x$	+	$72$ $72$
									+	$68$ $68$

Step 16

The final answer is the quotient plus the remainder over the divisor.

3 x^{2} + 7 x + 24 + \frac{68}{x - 3}

$3 x^{2} + 7 x + 24 + \frac{68}{x - 3}$

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